A360473 E.g.f. satisfies A(x) = exp( x * exp(x) * A(x)^2 ).

1, 1, 7, 82, 1441, 34036, 1013149, 36446698, 1538703457, 74607811048, 4086635087701, 249593193648646, 16819085803158577, 1239637405609740268, 99206330021667838285, 8567230421555333516746, 794104205843228382969409

Offset: 0

Author

Seiichi Manyama, Feb 08 2023

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..344
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A273954, A360465, A360474.

Programs

Mathematica

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x*Exp[x]*A[x]^2] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

PARI

a(n) = sum(k=0, n, k^(n-k)*(2*k+1)^(k-1)*binomial(n, k));

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x*exp(x))/2)))

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(-lambertw(-2*x*exp(x))/(2*x*exp(x)))))

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(sum(k=0, N, (k+1)^(k-1)*(2*x*exp(x))^k/k!))))

Formula

a(n) = Sum_{k=0..n} k^(n-k) * (2*k+1)^(k-1) * binomial(n,k).

E.g.f.: A(x) = exp( -LambertW(-2*x * exp(x))/2 ).

E.g.f.: A(x) = sqrt( -LambertW(-2*x * exp(x)) / (2*x * exp(x)) ).

E.g.f.: A(x) = sqrt( Sum_{k>=0} (k+1)^(k-1) * (2*x * exp(x))^k / k! ).

a(n) ~ sqrt(1 + LambertW(exp(-1)/2)) * n^(n-1) / (2 * exp(n - 1/2) * LambertW(exp(-1)/2)^n). - Vaclav Kotesovec, Feb 17 2023